Longitudinal data, characterized by repeated measurements on the same subjects over time, are ubiquitous in biomedical sciences, economics, social sciences, and engineering. Analyzing such data presents unique statistical and computational challenges, including within-subject correlation, time-varying covariates, irregular observation times, informative dropout, and high dimensionality. While traditional statistical methods, such as linear mixed-effects models and generalized estimating equations, remain foundational, they often struggle with complex nonlinear dynamics, ultra-high-dimensional feature spaces, and very large sample sizes. Over the past two decades, machine learning (ML) and artificial intelligence (AI) methods have emerged as powerful complementary approaches to address these limitations. This review provides a comprehensive survey of mathematical and computational methods for longitudinal data analysis. We cover classical statistical models, penalized regression techniques, tree-based ensemble methods, kernel machines, Bayesian hierarchical models, and modern deep learning architectures, including recurrent neural networks, temporal convolutional networks, attention-based Transformers, neural ordinary differential equations, and generative models. We propose a unified taxonomy that organizes existing methods along two primary axes: the underlying mathematical framework and the analytical objective. For each category, we present detailed mathematical formulations, discuss key theoretical properties, examine computational considerations, and summarize representative reported applications drawn from the published literature. To increase the practical value of this review, we provide a cross-cutting comparison of method families against five key challenges (within-subject correlation, irregular sampling, missing data, high dimensionality, and scalability) and offer concrete guidance on method selection according to sample size, dimensionality, and analytical objective. Finally, we critically evaluate the strengths and limitations of these approaches, with particular emphasis on interpretability, scalability, handling of missing data, robustness to covariance misspecification, and uncertainty quantification.

Evaluating whether clinical classifiers perform equitably across patient subgroups is a central requirement for the responsible deployment of machine learning in medicine. Conventional approaches test one fairness metric at a time, such as sensitivity, positive predictive value, or area under the receiver operating characteristic curve, and therefore cannot detect disparities that manifest only in the joint structure of a group-specific confusion matrix. We develop a unified hypothesis-testing framework rooted in random matrix theory that compares demographic groups through the $L^2$ distance between their joint eigenvalue densities, yielding a scalar spectral divergence that is sensitive to every cell of the $2\times 2$ confusion matrix simultaneously. We derive the closed-form spectral divergence for Gaussian-approximated eigenvalue densities, prove almost-sure consistency of the empirical estimator via the delta method, and construct an extreme-value (Gumbel) test statistic with family-wise error rate control. Monte Carlo experiments comprising $\mathrm{10,000}$ replications across balanced, moderately imbalanced, and severely imbalanced group-size regimes show that the spectral test keeps Type I errors close to its nominal level while achieving power exceeding $90\%$ in complex and multi-dimensional violation scenarios, where the best single-metric competitor reaches at most $63\%$. Three clinical benchmark datasets from the UCI Machine Learning Repository utilised include Pima Indians Diabetes ($n=768$), Cleveland Heart Disease ($n=303$), and Heart Failure Clinical Records ($n=299$). Results confirm that the spectral method detects statistically significant ($p<0.001$) performance disparities missed by all three conventional tests. These results support eigenvalue-based divergence as a practical, model-agnostic diagnostic equity tool for clinical machine learning audits.